Step |
Hyp |
Ref |
Expression |
1 |
|
r1pval.e |
⊢ 𝐸 = ( rem1p ‘ 𝑅 ) |
2 |
|
r1pval.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
r1pval.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
r1pval.q |
⊢ 𝑄 = ( quot1p ‘ 𝑅 ) |
5 |
|
r1pval.t |
⊢ · = ( .r ‘ 𝑃 ) |
6 |
|
r1pval.m |
⊢ − = ( -g ‘ 𝑃 ) |
7 |
2 3
|
elbasfv |
⊢ ( 𝐹 ∈ 𝐵 → 𝑅 ∈ V ) |
8 |
7
|
adantr |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝑅 ∈ V ) |
9 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = ( Poly1 ‘ 𝑅 ) ) |
10 |
9 2
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = 𝑃 ) |
11 |
10
|
fveq2d |
⊢ ( 𝑟 = 𝑅 → ( Base ‘ ( Poly1 ‘ 𝑟 ) ) = ( Base ‘ 𝑃 ) ) |
12 |
11 3
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( Base ‘ ( Poly1 ‘ 𝑟 ) ) = 𝐵 ) |
13 |
12
|
csbeq1d |
⊢ ( 𝑟 = 𝑅 → ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ⦋ 𝐵 / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) ) |
14 |
3
|
fvexi |
⊢ 𝐵 ∈ V |
15 |
14
|
a1i |
⊢ ( 𝑟 = 𝑅 → 𝐵 ∈ V ) |
16 |
|
simpr |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 ) |
17 |
10
|
fveq2d |
⊢ ( 𝑟 = 𝑅 → ( -g ‘ ( Poly1 ‘ 𝑟 ) ) = ( -g ‘ 𝑃 ) ) |
18 |
17 6
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( -g ‘ ( Poly1 ‘ 𝑟 ) ) = − ) |
19 |
|
eqidd |
⊢ ( 𝑟 = 𝑅 → 𝑓 = 𝑓 ) |
20 |
10
|
fveq2d |
⊢ ( 𝑟 = 𝑅 → ( .r ‘ ( Poly1 ‘ 𝑟 ) ) = ( .r ‘ 𝑃 ) ) |
21 |
20 5
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( .r ‘ ( Poly1 ‘ 𝑟 ) ) = · ) |
22 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( quot1p ‘ 𝑟 ) = ( quot1p ‘ 𝑅 ) ) |
23 |
22 4
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( quot1p ‘ 𝑟 ) = 𝑄 ) |
24 |
23
|
oveqd |
⊢ ( 𝑟 = 𝑅 → ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) = ( 𝑓 𝑄 𝑔 ) ) |
25 |
|
eqidd |
⊢ ( 𝑟 = 𝑅 → 𝑔 = 𝑔 ) |
26 |
21 24 25
|
oveq123d |
⊢ ( 𝑟 = 𝑅 → ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) = ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) |
27 |
18 19 26
|
oveq123d |
⊢ ( 𝑟 = 𝑅 → ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) = ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) = ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) |
29 |
16 16 28
|
mpoeq123dv |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
30 |
15 29
|
csbied |
⊢ ( 𝑟 = 𝑅 → ⦋ 𝐵 / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
31 |
13 30
|
eqtrd |
⊢ ( 𝑟 = 𝑅 → ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
32 |
|
df-r1p |
⊢ rem1p = ( 𝑟 ∈ V ↦ ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) ) |
33 |
14 14
|
mpoex |
⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ∈ V |
34 |
31 32 33
|
fvmpt |
⊢ ( 𝑅 ∈ V → ( rem1p ‘ 𝑅 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
35 |
1 34
|
syl5eq |
⊢ ( 𝑅 ∈ V → 𝐸 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
36 |
8 35
|
syl |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐸 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ) |
37 |
|
simpl |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → 𝑓 = 𝐹 ) |
38 |
|
oveq12 |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑓 𝑄 𝑔 ) = ( 𝐹 𝑄 𝐺 ) ) |
39 |
|
simpr |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → 𝑔 = 𝐺 ) |
40 |
38 39
|
oveq12d |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) |
41 |
37 40
|
oveq12d |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) ) |
42 |
41
|
adantl |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) ) |
43 |
|
simpl |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐹 ∈ 𝐵 ) |
44 |
|
simpr |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐺 ∈ 𝐵 ) |
45 |
|
ovexd |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) ∈ V ) |
46 |
36 42 43 44 45
|
ovmpod |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) ) |